Finding Area Between y = x² and y = √x

Calculus 2 · Axiom Academy

EXAMPLE Finding Area Between y = x^2 and Calculate the area enclosed by y = x^2 and using a definite integral. Find the area of the region bounded by the curves y = x^2 and . On the interval [0, 1], the curve lies above y = x^2 . The shaded region is the area we want. Nice work — you found the area between two curves end to end. Here is the playbook you just used: Find the intersection points: set the functions equal and solve to get the limits of integration, x = 0 and x = 1 . Decide which curve is on top: on [0, 1] we checked a test point and found , so is the upper curve. Integrate with the power rule: rewrite as x^ 1/2 to get the antiderivative . This same recipe handles any area-between-curves problem: find the intersections, decide which curve is on top, integrate (top − bottom), and evaluate. The one thing to never skip is subtracting in the correct order.

This is the written version of the interactive lesson above. See the full Calculus 2 course.